Donor Umbilical Cord Blood Transplant in Treating Patients With Advanced Hematologic Cancer

DISEASE CHARACTERISTICS: Diagnosis of 1 of the following advanced hematologic malignancies: Acute myeloid leukemia (AML) meeting the following criteria: Not expected to be curable with chemotherapy and meets ≥ 1 of the following criteria: High-risk cytogenetics (-7, -7q, -5, -5q, t[6,9], t[9,11], complex, Philadelphia chromosome positive [Ph+]) AML evolved from prior myelodysplasia AML secondary to prior chemotherapy Failed to achieve remission In second or subsequent remission Marrow blasts ≤ 10% (may be achieved using chemotherapy) Myelodysplastic syndromes (MDS) with high-risk features International Prognostic Scoring System (IPSS) score intermediate -2 or high-risk Marrow blasts ≤ 20% (may be achieved using chemotherapy) Acute lymphoblastic leukemia meeting the following criteria: Not expected to be curable with chemotherapy and meets ≥ 1 of the following criteria: High-risk cytogenetics (Ph+, t[4,11], 11q23 abnormalities, and monosomy 7) Required > 1 induction course to achieve remission Failed to achieve remission In second or subsequent remission Marrow blasts ≤ 10% (may be achieved using chemotherapy) Chronic myelogenous leukemia meeting ≥ 1 of the following criteria: Accelerated phase Chronic phase refractory to imatinib mesylate Blastic phase Marrow blasts ≤ 10% (may be achieved using chemotherapy) Multiple myeloma meeting 1 of the following criteria: Stage II or III disease with > first relapse or refractory disease Newly diagnosed disease with chromosome 13 abnormalities Lymphoma meeting the following criteria: One of the following subtypes: Diffuse large cell lymphoma Mantle cell lymphoma Peripheral T-cell lymphoma T-natural killer (NK) cell lymphoma Hodgkin's lymphoma Disease failed to respond to primary therapy, progressed, or recurred after prior therapy Patients who have failed autologous stem cell transplantation are eligible provided it has been > 1 year since transplant No rapid progression of malignant disease Not eligible for autologous stem cell transplantation Available umbilical cord blood (1-3 units) donor matching at ≥ 4 of 6 HLA antigens (A, B, and DR) Patients with an HLA-identical or 1 antigen-mismatched related donor OR a potential HLA-matched unrelated donor matching at > 6/8 (A, B, C, DR) alleles are not eligible PATIENT CHARACTERISTICS: ECOG performance status 0-2 Creatinine < 2.0 mg/dL Creatinine clearance > 40 mL/min Bilirubin < 2.0 mg/dL AST and alkaline phosphatase < 3 times upper limit of normal Hepatitis C and active hepatitis B allowed if patient has ≤ grade 2 inflammation or fibrosis by liver biopsy Ejection fraction > 40% by echocardiogram or MUGA DLCO > 40% of predicted Not pregnant or nursing Negative pregnancy test No known HIV infection No active infection requiring ongoing antibiotic treatment PRIOR CONCURRENT THERAPY: See Disease Characteristics